Timeline for Examples of sequences whose asymptotics can't be described by elementary functions
Current License: CC BY-SA 2.5
4 events
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| Nov 9, 2010 at 6:39 | comment | added | Denis Serre | What unifies $\tau$ and the examples I gave is that they are multiplicative functions: if $n\wedge m=1$, then $f(nm)=f(n)f(m)$. Such functions are unlikely to have an asymptotics, unless each restriction $f_p$ of $f$ to $\{1,p,p^2,p^3,\ldots\}$ have the same asymptotics. | |
| Nov 8, 2010 at 23:00 | comment | added | Matt Young | In a similar vein, one can consider things like the number of equivalent classes of positive definite binary quadratic forms of discriminant $d$. Another example would be $\tau(n)$, the Fourier coefficients of the Ramanujan $\Delta$ function; if you prefer positive numbers, then take $|\tau(n)|^2$. Perhaps the best example is simply the characteristic function of the primes. We tend to think of these sequences as "random" so they don't have asymptotics with elementary functions. | |
| Nov 8, 2010 at 13:58 | comment | added | Qiaochu Yuan | Good point. Perhaps I should impose some monotonicity hypotheses... | |
| Nov 8, 2010 at 13:45 | history | answered | Denis Serre | CC BY-SA 2.5 |